The use of étale predates SGA, and "spread out" fits Grothendieck’s idea of all-encompassing topos, "vast" and "slack", better than usual, as these things go. The name of étale morphisms derives from that. Also, French étaler comes from old French estal, which meant position/place, same as Greek topos, although it is unclear if that was intended too. See What are the benefits of viewing a sheaf from the “espace étalé” perspective? thread on Math Overflow for a more mathematical discussion.
Here is Grothendieck’s own explanation from Récoltes et Semailles:
"The crucial thing here, from the viewpoint of the Weil conjectures, is that the new notion [of space] is vast enough, that we can associate to each scheme a “generalized space” or “topos” (called the “étale topos” of the scheme in question). Certain “cohomology invariants” of this topos (“childish” in their simplicity!) seemed to have a good chance of offering “what it takes” to give the conjectures their full meaning, and (who knows!) perhaps to give the
means of proving them."
The idea was part of Grothendieck's general strategy for proving the Weil conjectures, which Serre explained to him in cohomological terms in 1955. Étale covers were inspired by Serre's "isotrivial covers". As McLarty's comments in The Rising Sea:
"Cohomology gives algebraic invariants of a topos, just as it used to give invariants of a topological space. Each topological space determines a topos with the sheaf cohomology. Each group determines a topos with the group cohomology. The same, Grothendieck knew, would work for cases yet unimagined... For the Weil conjectures it only remained to find the natural topos for each arithmetic space—recalling that up to 1956 or so the spaces themselves were not adequately defined. In fact this conception of “toposes” came to Grothendieck as the way to combine his theory of schemes with Serre’s idea of isotrivial covers and produce the cohomology."
This strategy was put in motion in collaboration with Artin in the early 1960-s, the time of SGA, see Jackson's As If Summoned from the Void:
"When Grothendieck came to Harvard in 1961, “I asked him to tell me the definition of étale cohomology,” Artin recalled with a laugh. The definition had not yet been formulated precisely. Said Artin, “Actually we argued about the
definition for the whole fall”. After moving to the Massachusetts Institute of
Technology in 1962, Artin gave a seminar on étale cohomology. He spent much of the following two years at the IHÉS working with Grothendieck."